Question

Hydrogen can cause fire hazards, and hydrogen gas leaking into surrounding air can lead to spontaneous ignition with extremely hot flames. Even at very low leakage rate, hydrogen can sustain combustion causing extended fire damages. Hydrogen gas is lighter than air, so if a leakage occurs it accumulates under roofs and forms explosive hazards. To prevent such hazards, buildings containing source of hydrogen must have adequate ventilation system and hydrogen sensors. Consider a metal spherical vessel, with an inner diameter of \(5 \mathrm{~m}\) and a thickness of \(3 \mathrm{~mm}\), containing hydrogen gas at \(2000 \mathrm{kPa}\). The vessel is situated in a room with atmospheric air at \(1 \mathrm{~atm}\). The ventilation system for the room is capable of keeping the air fresh, provided that the rate of hydrogen leakage is below \(5 \mu \mathrm{g} / \mathrm{s}\). If the diffusion coefficient and solubility of hydrogen \(\mathrm{gas}\) in the metal vessel are \(1.5 \times 10^{-12} \mathrm{~m}^{2} / \mathrm{s}\) and \(0.005 \mathrm{kmol} / \mathrm{m}^{3}\).bar, respectively, determine whether or not the vessel is safely containing the hydrogen gas.

Short answer

Answer: Yes, the vessel is safely containing the hydrogen gas because the calculated leakage rate (0.78 µg/s) is less than the allowable leakage rate (5 µg/s).

Step-by-step solution

Calculate surface area of the sphere

First, we need to calculate the surface area of the sphere. The formula for the surface area (A) of a sphere is given by \(A = 4 \pi r^2\), where r is the radius of the sphere: The radius of the sphere \(r\): \(r = \frac{Diameter}{2} = \frac{5~m}{2} = 2.5~m\) \(A = 4 \pi (2.5~m)^2 = 4 \pi (6.25~m^2) \approx 78.54~m^2\)

Calculate the concentration difference

Next, we need to calculate the concentration difference (ΔC) by converting the pressure difference to a concentration difference using the solubility (S) factor: \(\Delta P = P_{outside} - P_{inside} = 1~atm - (2000~kPa \times \frac{1~atm}{101.325~kPa}) \approx -19.82~atm\) \(\Delta C = S \times \Delta P = 0.005~\frac{kmol}{m^3.bar} \times (-19.82~bar) \approx -0.0991~\frac{kmol}{m^3}\) Note that the concentration difference is negative because the hydrogen gas concentration is higher inside the vessel than outside. This will cause the gas to diffuse from the inside to the outside.

Calculate the rate of diffusion

Next, we will use Fick's law of diffusion to calculate the rate of hydrogen leakage through the vessel walls: \(J = -D \times \frac{ΔC}{\delta}\), where J is the diffusion flux and \(\delta\) is the wall thickness. \(J = - (1.5 \times 10^{-12}~\frac{m^2}{s}) \times \frac{-0.0991~\frac{kmol}{m^3}}{3 \times 10^{-3}~m} \approx 4.95 \times 10^{-9}~\frac{kmol}{m^2.s}\)

Calculate the total leakage rate

Now, we need to calculate the total leakage rate (R) by multiplying the diffusion flux (J) by the surface area (A) of the sphere. Afterward, we need to convert the total leakage rate from kmol/s to µg/s. \(R = J × A = (4.95 \times 10^{-9}~\frac{kmol}{m^2.s}) × (78.54~m^2) \approx 3.89 \times 10^{-7}~\frac{kmol}{s}\) To convert the leakage rate to µg/s, we will use the molar mass (M) of hydrogen. The molar mass of hydrogen is 2 ~g/mol or 2000 ~µg/mmol. \(R_{µg/s} = R \times M = 3.89 \times 10^{-7}~\frac{kmol}{s} \times (2000~\frac{µg}{mmol}) \approx 0.78~\frac{µg}{s}\)

Compare the total leakage rate to the allowable leakage rate

Finally, we will compare the calculated total leakage rate (in µg/s) to the allowable leakage rate (5 µg/s): \(0.78~\frac{µg}{s} < 5~\frac{µg}{s}\) Since the calculated leakage rate is less than the allowable rate, the vessel is safely containing the hydrogen gas.

Key concepts

Fick's Law of Diffusion

Understanding how particles move from areas of high concentration to areas of low concentration is essential in fields like environmental engineering and materials science. This process is elegantly described by Fick's Law of Diffusion, which plays a crucial role in predicting the behavior of gas leaks, such as hydrogen.

Fick's First Law states that the rate of diffusion, or flux (\( J \)) of a substance is proportional to the negative gradient of the concentration (\( abla C \)). In simpler terms, it explains how fast a gas spreads out. The equation can be expressed as: \[ J = -D \times \frac{\Delta C}{\delta} \] where \( J \) is the diffusion flux (\( kmol/m^2.s \)), \( D \) is the diffusion coefficient (\( m^2/s \)), \( \Delta C \) is the concentration difference (\( kmol/m^3 \)), and \( \delta \) is the thickness of the barrier (\( m \)).

The negative sign indicates that diffusion occurs in the direction opposite to the increasing concentration gradient. In our exercise, Fick's Law helps us calculate whether hydrogen gas leaks at a safe rate through the metal of the spherical vessel.

Spherical Vessel Surface Area Calculation

When dealing with problems involving gases and containers, it is often necessary to calculate the container's surface area. This is particularly true for spherical vessels, which have a unique surface area calculation due to their shape.

To calculate the surface area of a sphere, you use the formula: \[ A = 4 \pi r^2 \] where \( A \) is the surface area and \( r \) is the radius of the sphere. Calculating the surface area allows us to quantify the total area through which gas can diffuse. In our example involving hydrogen gas, we need to know the surface area of the vessel to determine the total hydrogen leakage rate.

The diameter given in the exercise is converted to radius by dividing by two, and the area is then calculated by plugging the radius into the formula. This step is crucial as it serves as a multiplier for the diffusion flux in calculating the total leakage rate.

Concentration Difference Calculation

The driving force behind diffusion is the concentration difference between two regions. In the case of our spherical vessel containing hydrogen gas, we're concerned with the difference in hydrogen concentration between the inside of the vessel and the outside atmosphere.

The concentration difference (\( \Delta C \) is calculated using the gas pressures and solubility of hydrogen in the vessel material. The solubility product (\( S \) in \( kmol/m^3.bar \) gives us a direct way to convert pressure difference to concentration difference. Mathematically, it is given as: \[ \Delta C = S \times \Delta P \] where \( \Delta P \) is the pressure difference between the inside and outside of the vessel. It's essential to note that the negative value obtained for \( \Delta C \) indicates that the pressure is higher inside the vessel, hence causing the gas to diffuse outwards.

The concentration difference is the foundation for applying Fick's Law and ultimately determining the leakage rate of hydrogen gas. The calculated negative value of concentration difference signifies a natural diffusion from higher to lower concentration, a phenomenon that can lead to potential hazards if not properly managed.